Hunting for exotic doubly hiddencharm/bottom tetraquark states
Abstract
We develop a moment QCD sum rule method augmented by fundamental inequalities to study the existence of exotic doubly hiddencharm/bottom tetraquark states made of four heavy quarks. Using the compact diquarkantidiquark configuration, we calculate the mass spectra of these tetraquark states. There are 18 hiddencharm tetraquark currents with , , , , , , , and . We use them to perform QCD sum rule analyses, and the obtained masses are all higher than the spontaneous dissociation thresholds of two charmonium mesons, which are thus their dominant decay modes. The masses of the corresponding hiddenbottom tetraquarks are all below or very close to the thresholds of the and , except one current of . Hence, we suggest to search for the doubly hiddencharm states in the and channels.
keywords:
Tetraquark states, QCD sum rules, Moment method PACS: 12.39.Mk, 12.38.Lg, 11.40.qIn QED there exist the multilepton bound states composed of and Hylleraas and Ore (1947); Cassidy and Mills (2007). While, in QCD the scalar mesons below 1 GeV may have the flavor configurations and Jaffe (1977); Chen et al. (2007); Black:1998wt (), so are possible multiquark states. Actually, at the birth of the quark model, these multiquark states beyond conventional mesons and baryons were proposed by GellMann GellMann (1964) and Zweig Zweig (1964). Searching for multiquark matter has been an extremely intriguing research issue in the past fifty years, since it provides important hints to deepen our understanding of the nonperturbative QCD Chen:2016qju (). With experimental progress, more and more candidates of multiquark matter were reported over the past decade, including dozens of charmoniumlike states Olive et al. (2014), the hiddencharm pentaquarks and Aaij et al. (2015), and the observed by the D0 Collaboration D0:2016mwd () but not confirmed in the LHCb Aaij et al. (2016) and CMS Collaboration (2016) experiments. Hence, we conclude now to be the suitable time to hunt for more exotic multiquark states.
When studying multiquark matter, distinguishing these possible multiquark states from the conventional hadrons is a crucial task to establish their exotic nature. An explicit road map to achieve this aim is to study those exotic multiquark states with masses far away from the mass range of the observed conventional hadrons. We note that the pairs were observed in LHCb Aaij et al. (2012), DØ Abazov et al. (2014) and CMS Khachatryan et al. (2014); the simultaneous events were recently observed in DØ Abazov et al. (2016) and CMS CMS1 (); the simultaneous events were also recently observed in CMS Khachatryan et al. (2017). All these events may be used to investigate the fourheavyquark () states, which fully satisfy this condition, and so are very good multiquark candidates.
In this letter we develop a moment QCD sum rule method augmented by fundamental inequalities to systematically study the doubly hiddencharm/bottom tetraquark states and with quantum numbers , , , , , , , and . In our study, we find a typical common peculiarity of the predicted masses of these doubly hiddencharm and hiddenbottom tetraquark states, i.e., the masses of all tetraquarks are higher than the two charmonium thresholds while the tetraquark states (except one state of ) lie below the dissociation thresholds of two bottomonium states, and these masses do not overlap with the mass ranges of the observed charmonia/bottomonia. In addition to perform the mass spectrum analysis, their OZIallowed strong decay behaviors are also predicted in the present work, which are useful for future experimental searching.
These doubly hiddencharm/bottom tetraquark states did not receive much attention because of the absence of experimental data, unlike the (singly) hiddencharm/bottom tetraquark system, where represents the light quark and the heavy quark Chen and Zhu (2010, 2011); Chen et al. (2015b). However, with the running of LHC at 13 TeV and the forthcoming BelleII, searching for these doubly hiddencharm and hiddenbottom tetraquark states will probably become the next potential experimental issue in the near future. Systematic theoretical studies of the doubly hiddencharm/bottom tetraquark states seem to be imperative and useful for future experimental studies. However, the light mesons (, , , , ) can not be exchanged between the two charmonium/bottomonium states, which limits the use of many phenomenological methods Chen:2016qju (). To study these doubly hiddencharm/bottom tetraquark states, our present calculations in QCD sum rules and a very recent work in the framework of the colormagnetic interaction Wu:2016vtq () are both crucial to shed light on the longstanding debate on whether there exist such tetraquark states Iwasaki (1975); Chao (1981); Ader et al. (1982); Ballot and Richard (1983); Heller and Tjon (1985); Lloyd and Vary (2004); SilvestreBrac (1992); SilvestreBrac and Semay (1993); Barnea et al. (2006); Berezhnoy et al. (2012). We note that the traditional SVZ QCD sum rules Shifman et al. (1979a); Reinders et al. (1985); Colangelo and Khodjamirian (2000), which we used to study the (singly) hidden charm pentaquarks and Chen et al. (2015), can not be easily applied to study these doubly hiddencharm/bottom states, because of the four heavy quarks. It is another version of QCD sum rules, the moment QCD sum rules Shifman et al. (1979a); Reinders et al. (1985); Colangelo and Khodjamirian (2000), which we find to be capable of dealing with these states. The moment sum rules have been very successfully used for reproducing the charmonium and bottomonium mass spectra at the point Shifman et al. (1979a, a); 1983Nikolaevp526526 () and Reinders et al. (1981, 1985) and determining the strong coupling constant and the heavy quark masses 1997Jaminp334352 (); 2001Eidemullerp203210 (); 2001Kuhnp588602 (). Therefore, we shall use the moment QCD sum rules method in this letter to systematically study the doubly hiddencharm/bottom states as the first theoretical calculation for such systems in QCD.
To start our study, we first construct local (no derivative operators) tetraquark interpolating currents with four heavy quarks and in the diquarkantidiquark configuration. Consisting of the same heavy quarks, the flavor structures for both the diquark and antidiquark are symmetric. The color structures are then fixed by the Fermi statistics for various Lorentz bilinear operators, i.e., symmetric for the diquark fields , , , and antisymmetric for , and , where are color indices. Following Refs. Du et al. (2013); Chen (2013), we obtain the color singlet tetraquark operators with four heavy quark fields and definite quantum numbers outlined in the following, where the colour structure can be either symmetric or antisymmetric .
The interpolating currents with are
(1) 
The interpolating currents with and are
(2) 
in which and couple to the states with , and couples to the states with . The interpolating currents with and are
(3)  
in which and couple to the states with , and and couple to the states with . The interpolating currents with and are
(4) 
in which and couple to the states with , and and couple to the states with . The interpolating current with is
(5)  
In the following, we will explore the doubly hiddencharm/bottom tetraquark systems in the framework of moment QCD sum rules. We start with the twopoint correlation functions
(6) 
in which the interpolating currents , and couple to the scalar, vector and tensor states via the following relations respectively:
(7) 
where , and are coupling constants between interpolating currents and hadron states. and are the polarization vector and tensor, respectively.
The twopoint correlation function can be expressed in the form of the dispersion relation at the hadron level
(8) 
where is the mass of the physical state and are unknown subtraction constants. The imaginary part of the correlation function can be written as a sum over functions,
(9) 
in which a narrow resonance approximation is adopted since the lowest resonance is usually very sharp. The spectral function can thus be parameterized as a single sharp pole plus higher excited states and continuum. All the hadrons carrying the same quantum numbers as the current will give contributions to in Eq. (9). To pick out the lowest lying resonance in a particular channel, we define moments by taking derivatives of the correlation function in Euclidean region Reinders et al. (1985); Chen et al. (2014):
(10) 
where the spectral function . Using Eq. (9), we can write the moments as
(11) 
in which contains the contributions of higher states and the continuum. The behaviour of depends on both and . For a certain value of , tends to zero as goes to infinity, but this convergence becomes slower for larger . To eliminate in Eq. (11), we consider the following ratio of the moments
(12) 
For sufficiently large , one expects that for convergence Reinders et al. (1985). Then we can immediately extract the hadron mass of the lowest lying resonance from the above ratio
(13) 
The twopoint correlation function and moments can also be evaluated at the quarkgluon level using the operator product expansion (OPE) method. For these doubly hiddencharm/bottom systems, all QCD condensates involving the heavy quark fields do not contribute to , and we only need to calculate the perturbative term and gluon condensate in the OPE series. The full expressions of the correlation function and moments are too lengthy to be shown here. We only list the result for the interpolating current with as an example
(14)  
where .
To perform the numerical analysis, we use the following parameter values of the heavy quark masses (“running” masses in the scheme) and the gluon condensate Olive et al. (2014); Narison (1996); Ioffe and Zyablyuk (2003); Narison (2010)
(15)  
Using these parameter values, we can perform the numerical analysis of the and systems. There are two important parameters in the ratio defined in Eq. (12): and . The original literature used the moment at to perform moment method analysis Shifman et al. (1979a, a). However, it was emphasized that the higher dimensional condensates in the OPE series give large contribution to the correlation function and moments at the point Reinders et al. (1981, 1985), and thus lead to bad OPE convergence and unreliable mass predictions. In this letter, we use to ensure good OPE convergence. Thus, we need to carefully choose the two parameters and . For simplicity, we define and for and , respectively. These two parameters, and , are correlated with each other:

As mentioned above, a larger value of means slower convergence of . Then it will be difficult to extract the mass of the lowest lying resonance in Eq. (12). This can be compensated by taking higher derivative of for the lowest lying resonance to dominate.

However, a large value of means moving further away from the asymptotically free region. The OPE convergence would also become bad.
In this letter, we use the interpolating current with as an example to illustrate our calculations and analyses for the doubly hiddenbottom system. The upper limit of can be obtained by studying the convergence of the OPE series. Requiring the perturbative term to be larger than the gluon condensate term, we obtain upper limits for , respectively. The OPE fails for . The upper bound increases with respect to the value of . In principle, one can arrive at a region in the plane where the lowest lying resonance dominates the moments in Eq. (10) and the OPE series has good convergence. We will perform our analysis in such parameter regions.
The hadron mass in Eq. (13) is extracted as a function of and . In Fig. 1, we show the variation of the hadron mass with respect to for . The mass curves have plateaus which provide stable mass predictions. We find that these plateaus can be obtained by studying the integral expression of the moments in Eq. (10), which satisfies the Schwarz inequality Steele et al. (1999) (as a special case of Hölder’s inequality) in the following relation
(16) 
in which . In Fig. 2, we show the ratio as a function of and , in which in the blue region while in the pink region. The demarcation line between these two regions gives the values of for the plateaus in the mass curves. We thus obtain the plateaus for the mass curves in Fig. 1. One notes that the inequalities in Eq. (16) provide much stronger constrains for the plane than the OPE convergence. Finally, we obtain the hadron mass
(17) 
Currents  (GeV)  (GeV)  

Using the interpolating currents in Eqs. (2)–(5), we investigate all these channels and collect the masses of these and tetraquark states in Table 1. The errors come from the uncertainties of , the heavy quark masses and the gluon condensate in Eq. (15). We find that the positive parity states are lighter than the negative parity states. Comparing these states to the twomeson mass thresholds in Fig. 3, one notes that the tetraquark states lie below the mass thresholds of and , except one highest result for state. Although all tetraquarks lie above the twocharmonium thresholds, some of the positive parity states () are very close to the mass of the . These results for the tetraquarks are compatible with those predicted in Ref. Iwasaki (1975) and by a quarkgluon model Chao (1981), potential model Ader et al. (1982) and the parametrized Hamiltonian approach Lloyd and Vary (2004). In the framework of diquark model, the masses of the and tetraquarks with were also obtained by solving nonrelativistic Schroedinger equation Berezhnoy et al. (2012). However, their masses for the states are slightly lower than our results while a bit higher for the tetraquarks.
In general, the states mainly decay into two meson final states by spontaneous dissociation, like the decay properties for the light scalar mesons Rupp et al. (2016). For the doubly hiddencharm tetraquarks, they can decay via the spontaneous dissociation mechanism since they lie above the twocharmonium thresholds. For the doubly hiddenbottom states, they lie below any twobottomonium threshold and thus can not decay into two bottomonium mesons. Considering the restrictions of the symmetries, we collect their possible dissociation decay channels in Table 2 for both wave and wave.
Swave  Pwave  

,  ,  
,  
, ,  
, ,  
,  
, ,  
These tetraquarks can also decay into a doubly charmed/bottomed baryon () pair by the creation of a light quark pair so long as the kinematics allows. However, the thresholds of two doubly charmed/bottomed baryons are higher than the extracted masses of the and states in Table 1 Mattson et al. (2002). Hence, such decays are forbidden by the kinematics. The two singly heavy meson pair decay modes are possible via a heavy quark pair annihilation and a light quark pair creation at the same time. In general, these decays should be suppressed by the annihilation of a heavy quark pair. However, such suppression can be compensated by the large phase space factor Chao (1981). Consequently, the two heavy meson decay modes will contribute significantly to the total width of the doubly hiddencharm/bottom tetraquark states.
The doubly hiddencharm and hiddenbottom tetraquark states lie in the energy region much higher than the conventional charmonium and bottomonium mesons, respectively. They can be clearly distinguished experimentally from the normal states. The production of the states is extremely difficult because that requires two heavy quark pairs be created. However, the recent observations of the pair Aaij et al. (2012); Khachatryan et al. (2014) and the simultaneous Abazov et al. (2016); CMS1 () and events Khachatryan et al. (2017) shed some light for the production of these doubly hiddencharm/bottom tetraquarks. Thus, the and channels can be good choices to search for the doubly hiddencharm states, and LHCb, D0, and CMS are potential and ideal platforms.
To end this paper, we suggest to search for the doubly hiddencharm states in the and channels. Our predictions will provide new possible phenomena in hadron colliders. In the near future, we hope that these doubly hiddencharm/bottom tetraquark states can be observed at facilities such as LHCb, D0, CMS, RHIC and the forthcoming BelleII, where many heavy quarks are produced.
This project is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the National Natural Science Foundation of China under Grants 11205011, No. 11475015, No. 11375024, No. 11222547, No. 11175073, and No. 11575008; the Ministry of Education of China (SRFDP under Grant No. 20120211110002 and the Fundamental Research Funds for the Central Universities). X.L. is also supported by the National Youth Topnotch Talent Support Program (ThousandsofTalents Scheme).
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